For a reliable simulation of systems subject to noise, it is necessary to characterize the noise properly and develop efficient algorithms.

In the first part of this talk, I will present a numerical technique to model and simulate multiple correlated random processes. The method finds the appropriate expansion for each correlated random process by generalizing the Karhunen-Loeve (K-L) expansion, in particular, by releasing the bi-orthogonal condition of the K-L expansion. I will address the convergence and computational efficiency, in addition to some explicit formulae and analytical results.

In the second part, I will present an adaptive reduced basis method that enables an efficient simulation of parameterized stochastic PDEs. The method is employed by using an adaptive ANOVA and probabilistic collocation method to automatically identify the important dimensions and appropriate resolution in each dimension. The effectiveness of the method is demonstrated in anisotropic high-dimensional stochastic PDEs.

We provide a combinatorial characterization of all testable properties of k-graphs (i.e. k-uniform hypergraphs). Here, a k-graph property 𝒫 is testable if there is a randomized algorithm which quickly distinguishes with high probability between k-graphs that satisfy 𝒫 and those that are far from satisfying 𝒫. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. This is joint work with Felix Joos, Deryk Osthus and Daniela Kühn.

Based on the quantum white noise theory, we introduce the new concept of quantum white noise derivatives of white noise operators. As applications we solve implementation problems for the canonical commutation relation and for a quantum extension of Girsanov transformation.

In the second lecture we continue the discussion of orthogonal polynomials, now dealing with multi-variable functions. By introducing creation, annihilation, and preservation operators for the multi-variables, we construct again an interacting Fock space (IFS). Thereby we extend the theory of orthogonal polynomials in the 1-dimensional space to that in the multi-dimensional space. As a byproduct we show the relationship between the support of the measure and the deficiency rank of the form generator, which appears in the construction of the IFS. We finish with some open problems. This lecture is based on the joint work with A. Dhahri (Chungbuk) and N. Obata (Tohoku).

We start with the standard construction of generalized white noise functionals as infinite dimensional distributions and we study the analytic characterization theorem for S-transform of generalized white noise functionals. Then we study basic concepts and results on white noise operators which is necessary for the study of quantum white noise calculus. The analytic characterization of operator symbols and the Fock expansion theorem are of particular importance.

I will discuss some aspects of the algebraic structure of finitely generated groups of diffeomorphisms of compact one-manifolds. In particular, we show that if G is not virtually metabelian then (G x Z)*Z cannot act faithfully by C^2 diffeomorphisms on a compact one-manifold. Among the consequences of this result is a completion of the classification of right-angled Artin groups which admit faithful C^{\infty} actions on the circle, a program initiated together with H. Baik and S. Kim. This represents joint work with S. Kim.

3-d printing gives us unprecedented ability to tailor microstructures to achieve desired goals. From the mechanics perspective one would like, for example, to know how to design structures that guide stress, in the same way that conducting fibers are good for guiding current. In that context the natural question is: what are the possible pairs of (average stress, average strain) that can exist in the material. A more grand question is: what are the possible effective elasticity tensors that can be achieved by structuring a material with known moduli. This is a highly non-trivial problem: in 3-dimensions elasticity tensors have 18 invariants and even an object as simple as a distorted hypercube in 18 dimensions requires about 4.7 million numbers to specify it. Here we review some of the progress that has been made on this question. This is joint work with Marc Briane and Davit Harutyunyan

Let P be a graph property. We look at graph colorings with k colors where each color class induces a graph satisfying P. By a result of Makowsky and Zilber (2005) the number of such colorings xP(G;k) is a polynomial in k. We present recent results and open problems on the complexity of evaluating xP(G;L) for various properties P and (not only integer) values of L.
This is joint work with A. Goodall, M. Hermann, T. Kotek and S. Noble which was initiated during last year’s program “Counting Complexity and Phase Transitions”. See also https://arxiv.org/abs/1701.06639

The behavior of shape memory materials, the response of composites, and inverse problems (where one seeks to determine what is inside a body from boundary measurements) would at first sight seem to have little in common. However there are unifying mathematical themes that underlie them all. Finding composites that have the best properties, for design applications, often reduces to a type of energy minimization problem with a non-linear energy, even though the underlying physical equations may be linear. The same sort of energy minimization problems govern the response of shape memory materials. Similarly, the response of inhomogenous bodies, as governed by the appropriately defined Dirichlet to Neumann map, is similar in many respects to an effective tensor in the theory of composites, and this connection can be made more mathematically explicit. As a result of these connections, mathematical tools developed in one area can be applied to problems in the other areas.

All finite graphs satisfy the two properties mentioned in the title. I will explain what I mean by this, and speculate on generalizations and interconnections. This talk will be non-technical: Nothing will be assumed beyond basic linear algebra.

In this talk I will introduce tensor networks for breaking the curse-of-dimensionality in large scale optimizaition problems. I will mainly focus on the tensor train (TT) format, which is one of the simplest tensor networks. I will show how large-scale optimization problems,

which are intractable by standard numerical methods, can be solved by using the concept of tensorization and TT format. In addition, I will discuss several state-of-the art numerical schemes for tensor networks including truncated iteration scheme, alternating linear scheme, and Riemannian optimization approach.

When network users are satisficing decision-makers, the resulting traffic pattern attains a satisficing user equilibrium, which may deviate from the (perfectly rational) user equilibrium. In a satisficing user equilibrium traffic pattern, the total system travel time can be worse than in the case of the PRUE. We show how bad the worst-case satisficing user equilibrium traffic pattern can be, compared to the perfectly rational user equilibrium. We call the ratio between the total system travel times of the two traffic patterns the price of satisficing, for which we provide an analytical bound. Using the sensitivity analysis for variational inequalities, we propose a numerical method to quantify the price of satisficing for any given network instance.

Let F be a family of convex sets in R^d coloured using d+1 colours. Lovasz’s Colourful Helly Theorem states that if any colourful subfamily of convex sets is intersecting, then one of the monochromatic families is intersecting. We study what happens with the rest of the families.

Approximation algorithms and fixed-parameter tractable (FPT) algorithms have been two major ways to deal with NP-hardness of combinatorial optimization problems. The notion of FPT approximation can be naturally defined, and it is getting significant attention recently. Starting from gentle introductions to approximation algorithms and FPT algorithms, I will talk about my three recent results on FPT approximability.– Given a graph G = (V, E) and an integer k, we study k-Vertex Separator, where the goal is to remove the minimum number of vertices such that each connected component in the resulting graph has at most k vertices. We give an O(log k)-FPT approximation algorithm for k-Vertex Separator. Our result improves the best previous graph partitioning algorithms.– We also study k-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. We present an O(log k)-FPT approximation algorithm for k-Path Transversal. There was no nontrivial approximation algorithm for k > 4 before this work.– Finally, k-cut is the problem where we want to remove the minimum number of edges such that the graph has at least k connected components. We give a (2 – ε)-FPT approximation algorithm for some epsilon > 0, improving upon a (non-FPT) 2-approximation.The third result is joint work with Anupam Gupta and Jason Li.

In a financially globalized world emerging market countries have different attributes from developed ones. Emerging countries may not be able to enjoy such benefits as consumption smoothing, efficient investment and risk diversification as developed countries do. One example is the sovereign wealth fund, a precautionary savings fund run by government. Instead, they may suffer so-called capital inflow problem caused by massive capital inflows followed by a sudden stop and reversal. The global financial crisis which hit Korea in 2008 is a classic example. The root source of the problem is that these countries, located in the periphery of the global financial world, cannot produce safe assets. The direct implication is that emerging market countries must import safe assets often financed by current account surplus. Furthermore, the central banks in emerging market countries should ultimately bear the risk of currency and maturity mismatches of domestic residents. That is, they should serve not only a lender of last resort but as an insurer of last resort, which makes central banking in emerging market countries complicated. Considering that capital flows are procylical and emerging countries are unable to produce safe assets capital flows to/from emerging market countries even strengthen the case for procylicality. Consequently, it is inevitable that emerging economies in the periphery are heavily influenced by the monetary policy of the center countries, and have essentially no room for independent monetary policy. In fact, the trilemma (inability to choose flexible exchange rates, capital mobility and independent monetary policy) in theory turns out a dilemma in reality. This seminar has two parts, central banking in emerging market and central banking in Korea. After reviewing central banking in emerging market in general I will discuss specific issues related to the monetary policy of the Bank of Korea.

The Langlands program consists of a huge web of tantalizing conjectures in many different directions, e.g., relating number theory, representation theory, algebraic geometry, and harmonic analysis in an unexpected way. After a leisurely introduction to the program through a brief historical review, we introduce yet another point of view on the program, inspired by work of physicists Kapustin and Witten, by investigating a certain 4-dimensional quantum field theory. Incidentally, this turns out to provide surprising new structures on the program. This talk, based on my thesis work, is mostly aimed at non-specialists.

The KdV equation is a nonlinear partial differential equation describing waves on shallow water surfaces. In spite of its nonlinearity, this is exactly solvable as it admits a surprisingly rich structure like infinite-dimensional symmetries, called the KdV hierarchy. From a completely different direction, for a Calabi-Yau manifold X, one can consider a generating function of certain enumerative invariants of X. Witten conjectured and Kontsevich proved the mysterious claim that when X is a point, the generating function is governed by the KdV hierarchy. I will explain a program toward understanding what happens when X is a general Calabi-Yau manifold. This talk is based on a joint project in progress with Weiqiang He and Si Li.

In a financially globalized world emerging market countries have different attributes from developed ones. Emerging countries may not be able to enjoy such benefits as consumption smoothing, efficient investment and risk diversification as developed countries do. One example is the sovereign wealth fund, a precautionary savings fund run by government. Instead, they may suffer so-called capital inflow problem caused by massive capital inflows followed by a sudden stop and reversal. The global financial crisis which hit Korea in 2008 is a classic example. The root source of the problem is that these countries, located in the periphery of the global financial world, cannot produce safe assets. The direct implication is that emerging market countries must import safe assets often financed by current account surplus. Furthermore, the central banks in emerging market countries should ultimately bear the risk of currency and maturity mismatches of domestic residents. That is, they should serve not only a lender of last resort but as an insurer of last resort, which makes central banking in emerging market countries complicated. Considering that capital flows are procylical and emerging countries are unable to produce safe assets capital flows to/from emerging market countries even strengthen the case for procylicality. Consequently, it is inevitable that emerging economies in the periphery are heavily influenced by the monetary policy of the center countries, and have essentially no room for independent monetary policy. In fact, the trilemma (inability to choose flexible exchange rates, capital mobility and independent monetary policy) in theory turns out a dilemma in reality. This seminar has two parts, central banking in emerging market and central banking in Korea. After reviewing central banking in emerging market in general I will discuss specific issues related to the monetary policy of the Bank of Korea.

In this talk, I will describe a project (work in progress) with Sang-Bum Yoo, on birational geometry of moduli spaces of parabolic bundles on the projective line in the framework of Mori’s program. It exhibits an interesting connection between representation theory and birational algebraic geometry via invariant theory.

Let M be a closed manifold that admits a nontrivial cover diffeomorphic to itself. Which manifolds have such a self-similar structure? Examples are provided by tori, in which case the covering is homotopic to a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is induced by a linear endomorphism of a torus on a quotient of the fundamental group. Under further hypotheses we show that a finite cover of M is a principal torus bundle. We use this to give an application to holomorphic self-covers of Kaehler manifolds.

Credit derivatives such as CDS and CDOs are financial instruments to hedge against the default risk. CDS is a kind of insurance that compensates for the loss of bonds issued by an individual company. CDOs issue a new security called tranche, which has various types of profit and risk structures. This lecture covers the structure and pricing of CDS and CDOs.

In order to measure the return or risk of an investment, it is necessary to reflect the correlation between the return of investment assets and the default random variables. Traditionally, correlation coefficients explain the dependency structure between elliptic distributions such as normal distribution well, but new dependency measures are needed because the actual data show a much different pattern from the elliptical distributions. Copulas are widely used as an alternative to these correlation coefficients and are often used in risk management and the valuation of credit derivatives such as CDOs. This lecture will cover the basic concepts of copula and its applications.

We establish the lower bound of the entropy of sums in prime cyclic groups and their applications. Main ingredients of our approach are extended rearrangement inequalities in prime cyclic groups and the function ordering in the sense of the majorization. Our entropy inequalities also have interesting applications on Discrete Entropy Power Inequalities, and the Littlewood-Offord problem. If time permits, we discuss how to establish the lower bound of the entropy of sums in integer through Sperner Theory.

Many basic PDE of physical interest, such as the three-dimensional Navier-Stokes equations, are "supercritical" in that the known conserved or bounded quantities for these equations allow the nonlinear components of the PDE to dominate the linear ones at fine scales. Because of this, almost none of the known methods for establishing global regularity for such equations can work, and global regularity for Navier-Stokes in particular is a notorious open problem. We present here some ways to show that if one allows some modifications to these supercritical PDE, one can in fact construct solutions that blow up in finite time (while still obeying conservation laws such as conservation of energy). This does not directly impact the global regularity question for the unmodified equations, but it does rule out some potential approaches to establish such regularity.

Let G be a graph on n vertices with independence number α. How large must a k-connected subgraph G contain? We shall present the best possible answers when α=2 and α=3. Some open questions will also be presented.

The discrepancy of a sequence f(1), f(2), ... of numbers is defined to be the largest value of |f(d) + f(2d) + ... + f(nd)| as n,d range over the natural numbers. In the 1930s, Erdos posed the question of whether any sequence consisting only of +1 and -1 could have bounded discrepancy. In 2010, the collaborative Polymath5 project showed (among other things) that the problem could be effectively reduced to a problem involving completely multiplicative sequences. Finally, using recent breakthroughs in the asymptotics of completely multiplicative sequences by Matomaki and Radziwill, as well as a surprising application of the Shannon entropy inequalities, the Erdos discrepancy problem was solved in 2015. In this talk I will discuss this solution and its connection to the Chowla and Elliott conjectures in number theory.

We proposes a computational framework for continuous time opinion dynamics with additive noise. We derive a non-local partial differential equation for the distribution of opinions differences. We use Mellin transforms to solve the stationary solution of this equation in closed form. This approach can be applied both to linear dynamics on an interaction graph and to bounded confidence dynamics in the Euclidean space. To the best of our knowledge, the closed form expression on the stationary distribution of the bounded confidence model is the first quantitative result on the equilibria of this class of models. The solutions are presented here in the simplest possible cases (small number of agents, present of stubborn agents, one dimensional opinions).

Hyperrings and hyperfields are algebraic structures which generalize commutative rings and fields. In this talk, we aim to introduce these exotic structures and also provide examples which illustrate how hyperrings and hyperfields show up in algebraic geometry and combinatorics following the idea of Baker and Bowler on `matroids over hyperfields'.

We introduce the concept of low rank-width colorings, generalizing the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez in [Grad and classes with bounded expansion I. Decompositions. EJC 2008]. We say that a class C of graphs admits low rank-width colorings if there exist functions N:ℕ→ℕ and Q:ℕ→ℕ such that for all p∈ℕ, every graph G∈C can be vertex colored with at most N(p) colors such that the union of any i≤p color classes induces a subgraph of rank-width at most Q(i).Graph classes admitting low rank-width colorings strictly generalize graph classes admitting low tree-depth colorings and graph classes of bounded rank-width. We prove that for every graph class C of bounded expansion and every positive integer r, the class {Gr: G∈C} of r-th powers of graphs from C, as well as the classes of unit interval graphs and bipartite permutation graphs admit low rank-width colorings. All of these classes have unbounded rank-width and do not admit low tree-depth colorings. We also show that the classes of interval graphs and permutation graphs do not admit low rank-width colorings. In this talk, we provide the color refinement technique necessary to show the first result. This is joint work with Sebastian Sierbertz and Michał Pilipczuk.

In this talk, we consider stochastic partial differential equations, especially, a parabolic Anderson model. This model shows intermittent phenomena, i.e., the solution becomes very big on small regions of different scales (we say tall peaks occur on small islands). We provide a way to quantify tall peaks and small islands by using the macroscopic fractal dimension theory by Barlow and Taylor. This is based on joint work with Davar Khoshnevisan and Yimin Xiao.

The interactions between particles and uid have received a bulk of attention due to a number of their applications in the eld of, for example, biotechnology, medicine, and in the study of sedimentation phenomenon, compressibility of droplets of the spray, cooling tower plumes, and diesel engines, etc. In this talk, we present coupled kinetic- uid equations. The proposed equations consist of Vlasov-Fokker-Planck equation with local alignment forces and the incompressible Navier-Stokes equations. For the equations, we establish the global existence of weak solutions, hydrodynamic limit, and large-time behavior of solutions. We also remark on blow-up of classical solutions in the whole space.

High-frequency financial data allow us to estimate large volatility matrices with relatively short time horizon. Many novel statistical methods have been introduced to address large volatility matrix estimation problems from a high-dimensional Ito process with micro-structure noise contamination. Their asymptotic theories require sub-Gaussian or some finite high-order moments assumptions. These assumptions are at odd with the heavy tail phenomenon that is pandemic in financial stock returns and new procedures are needed to mitigate the influence of heavy tails. In this paper, we introduce the Huber loss function with a diverging threshold to develop a robust realized volatility estimation. We show that it has the sub-Gaussian concentration around the conditional expected volatility with only finite fourth moments. With the proposed robust estimator as input, we further regularize it by using the principal orthogonal component thresholding (POET) procedure to estimate the large volatility matrix that admits an approximate factor structure. We establish the asymptotic theories for such low-rank plus sparse matrices. The simulation study is conducted to check the finite sample performance of the proposed estimation methods.

The circle is the only connected closed 1-dimensional manifold, and maybe that's why it has so many interesting features. In this talk, we would like to emphasize that there are many things we still do not know about this one of the simplest manifolds. We will survey many interesting recent results around the circle in the context of low-dimensional topology.